System and method for determining characteristic parameters in an aircraft

ABSTRACT

The present invention provides a method and a calculation system for an aircraft, with at least one sensor for detecting aeroelastic and flight-mechanical momenta of the aircraft, for detecting positions and movements of control surfaces of the aircraft or for detecting speeds of gusts of wind acting on the aircraft and comprising a calculation unit which calculates characteristic quantities of passenger comfort and cabin safety as well as momenta of the aircraft as a function of the sensor data provided by the sensors and a non-linear simulation model of the aircraft.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of PCT/EP2009/056659 filed May 29,2009 and claims the benefit of German Patent Application No. 10 2008 002124.5 filed May 30, 2008, and U.S. Provisional Application Ser. No.61/130,375 filed May 30, 2008, the entire disclosures of which areherein incorporated by reference.

FIELD OF THE INVENTION

The invention relates to a calculation system for an aircraft and to amethod for determining characteristic quantities and momenta of anaircraft.

Aircraft, such as aeroplanes or helicopters, are exposed to variousforces during flight. Significant influencing variables in this respectare the lifting forces generated by the aerofoils, the aerodynamicresistance of the aircraft, the weight or gravitational force acting ona centre of gravity of the aircraft, the shear force generated by theengines, the stick forces generated on the control surfaces of theaircraft and the torques caused by the respective forces. The massinertia of the aircraft or the mass inertia of the aircraft componentsalso plays a part in the above-mentioned forces. Flight maneuvers andair turbulence result in structural loads on the aircraft.

To predict the flight behaviour of an aircraft, systems of equations areused which are complex due to the large number of correlations betweenaeroelastic and flight-mechanical momenta. Conventional simulationsystems for simulating the behaviour of aircraft are based onsubstantially linear models of the structural dynamics of the stationaryand instationary aerodynamics, of aeroelastics, of the loads and theflight mechanics. In this respect, conventional systems of equationsconsider substantially linear characteristics of parameters.

The calculation accuracy of these conventional calculation systems usingsubstantially linear models is thus, however, relatively poor, i.e. theydo not reflect the actual behaviour of an aircraft in a sufficientlyaccurate manner.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to provide acalculation system and a method for determining characteristicquantities of an aircraft which accurately simulates the actualbehaviour of an aircraft.

This object is achieved by a calculation system proposed by the presentinvention.

The invention provides a calculation system for an aircraft with atleast one sensor for detecting aeroelastic and flight-mechanical momentaof the aircraft, for detecting positions and movements of controlsurfaces of the aircraft or for detecting speeds of gusts of wind actingon the aircraft; and with a calculation unit which calculatescharacteristic quantities of passenger comfort and cabin safety as wellas momenta of the aircraft as a function of the sensor data provided bythe sensors and a non-linear simulation model of the aircraft.

In an embodiment of the calculation system according to the invention,the calculation unit automatically adapts the non-linear simulationmodel using the sensor data provided by the sensors.

In an embodiment of the calculation system according to the invention,sensors are provided for detecting momenta of an on-board system of theaircraft.

In an embodiment of the calculation system according to the invention,the on-board system has at least one movable mass for damping anassociated part of the aircraft.

In an embodiment of the calculation system according to the invention,sensors for detecting flight-mechanical momenta of the aircraft alsomeasure deformations of parts of the aircraft.

In an embodiment of the calculation system according to the invention,the sensors for detecting flight-mechanical momenta of the aircraft andfor detecting aeroelastic momenta of the aircraft have acceleration orpressure sensors.

In an embodiment of the calculation system according to the invention,the calculation unit is provided in the aircraft or the data from thesensors of the aircraft is received by the calculation unit via awireless air interface.

In an embodiment of the calculation system according to the invention,the linear simulation model of the aircraft can be read out of a memory.

In an embodiment of the calculation system according to the invention,the calculation unit is connected to an input unit for inputtingparameters of the simulation model of the aircraft.

In an embodiment of the calculation system according to the invention,the calculation unit is connected to an output unit for outputtingcharacteristic quantities and momenta.

In an embodiment of the calculation system according to the invention,the on-board system of the aircraft is automatically controlled as afunction of the characteristic quantities and momenta, calculated by thecalculation unit, for minimising load forces and vibrations.

In an embodiment of the calculation system according to the invention,the on-board system of the aircraft can be connected to and disconnectedfrom different frequency ranges.

In an embodiment of the calculation system according to the invention,various masses of the on-board system which are fitted to parts of theaircraft can be activated as a function of an adjustable operating modeof the on-board system.

The invention also provides a method for determining characteristicquantities of passenger comfort and momenta of an aircraft, comprisingthe following steps:

-   -   (a) detection of aeroelastic and flight-mechanical momenta of        the aircraft, of positions and movements of control surfaces of        the aircraft and of speeds of gusts of wind acting on the        aircraft, to generate sensor data; and    -   (b) calculation of the characteristic quantities of passenger        comfort and of momenta of the aircraft as a function of the        generated sensor data and a stored, non-linear simulation model        of the aircraft.

The invention also provides a computer program with program commands forimplementing a method for determining characteristic quantities ofpassenger comfort and momenta of an aircraft, which comprises thefollowing steps:

-   -   (a) detection of aeroelastic and flight-mechanical momenta of        the aircraft, of positions and movements of control surfaces of        the aircraft and of speeds of gusts of wind acting on the        aircraft, to generate sensor data; and    -   (b) calculation of the characteristic quantities of passenger        comfort and of momenta of the aircraft as a function of the        generated sensor data and a stored, non-linear simulation model        of the aircraft.

The invention also provides a data carrier which stores such a computerprogram.

In the following, preferred embodiments of the calculation systemaccording to the invention and of the method according to the inventionfor determining characteristic quantities of passenger comfort and ofmomenta of an aircraft will be described with reference to theaccompanying figures to illustrate features which are essential to theinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows coordinate systems of a non-linear simulation model of anaircraft used in the calculation system according to the invention;

FIG. 2 shows a block diagram of a possible embodiment of the calculationsystem according to the invention;

FIG. 3 shows a block diagram of a further embodiment of the calculationsystem according to the invention;

FIG. 4A, 4B show diagrams to illustrate the non-linear simulation modelof an aircraft on which the calculation system according to theinvention is based;

FIG. 5A-5E show special cases of the non-linear simulation model onwhich the calculation system according to the invention is based;

FIG. 6A, 6B show diagrams for representing examples of use of thecalculation system according to the invention for aircraft;

FIG. 7A, 7B show diagrams for representing further examples of use ofthe calculation system according to the invention for aircraft.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

As can be seen from FIG. 1, the movements of an aircraft can bedescribed by characteristic quantities. The flight mechanics describethe behavior of an aircraft which moves through the atmosphere by meansof aerodynamics. The flight mechanics describe the behaviour of theentire system or of the aircraft, the position, altitude and flightspeed of an aerodynamic vehicle being calculated at any point in time.This is effected using motion equations which form a system of equationsof coupled differential equations. Maneuvering loads and structuralloads arise on an aircraft as a result of flight maneuvers and airturbulence. Maneuvering loads can be described by non-linear motionequations and are based on databases which state aerodynamic forces.Particularly large aircraft must also consider the elastic deformationsof their structure in addition to non-linear movements.

The movement of a rigid aircraft can be described by parameters. In eachcase three of these variables are combined into a vector, describing the

$\begin{matrix}\begin{matrix}{\mspace{79mu}{{Position}\text{:}}} & {\overset{\rightarrow}{S} = \begin{bmatrix}x & y & z\end{bmatrix}^{T}}\end{matrix} & (1) \\\begin{matrix}{{Angular}\mspace{14mu}{position}\mspace{14mu}\left( {{Euler}\mspace{14mu}{angles}} \right)\text{:}} & {\overset{\rightarrow}{\Phi} = {\begin{bmatrix}\phi \\\theta \\\Psi\end{bmatrix}\mspace{14mu}\begin{matrix}{{Bank}\mspace{14mu}{angle}\mspace{14mu}\left( {{roll}\mspace{14mu}{angle}} \right)} \\{{Inclination}\mspace{14mu}{angle}} \\\left( {{pitch}\mspace{14mu}{angle}} \right) \\{{Heading}\mspace{14mu}\left( {{yaw}\mspace{14mu}{angle}} \right)}\end{matrix}}}\end{matrix} & (2) \\\begin{matrix}{\mspace{79mu}{{Speed}\text{:}}} & {\overset{\rightarrow}{V} = \begin{bmatrix}u & v & w\end{bmatrix}^{T}}\end{matrix} & (3) \\\begin{matrix}{\mspace{79mu}{{Angular}\mspace{14mu}{velocity}\text{:}}} & {\overset{\rightarrow}{\Omega} = {\begin{bmatrix}p \\q \\r\end{bmatrix}\mspace{14mu}\begin{matrix}{{Roll}\mspace{14mu}{rate}} \\{Pitch} \\{rate} \\{Yaw} \\{rate}\end{matrix}}}\end{matrix} & (4)\end{matrix}$

The causes of the movement are the forces acting on the aircraft,Weight: {right arrow over (G)}=[G _(x) G _(y) G _(z)]^(T)  (5)shear and aerodynamic forces as well as the moments thereof, theresultants of which are combined in the vectors:Force: {right arrow over (R)}=[XYZ] ^(T)  (6)Moment {right arrow over (Q)}=[LMN] ^(T)  (7)

A further important quantity is the specific force measured by theaccelerometers.{right arrow over (b)}=[b _(x) b _(y) b _(z)]  (8)

The specific force is an indication of the acceleration impression ofthe pilot according to magnitude and direction and is defined as theratio of the resulting external force to the mass of the aircraft.

To determine Newton's equations and the angular momentum equation, theaccelerations and speeds are measured with respect to an inertialsystem. The earth is used as the inertial system, an earth frame ofreference F_(E) being defined in which the z-axis is directed towardsthe geocentre. The x- and y-axes are selected such that a right-handedsystem is produced. The axis system can be oriented, for example towardsmagnetic north. When evaluating the angular momentum equation, it hasproved advantageous to do this in a body frame of reference F_(B) ¹,because then the inertia sensor is constant. There are variousapproaches for establishing the axes of the body frame of reference, theorigin in each case being in the centre of gravity C of the aircraft.The main axis system is disposed such that the x-axis is in thedirection of the longitudinal axis of the aircraft and the z-axis isdirected vertically downwards relative thereto. Cy is selected such thata right-handed system is produced. If the stability axes are selected,the x-axis is disposed in the direction of the flight speed. The twoother axes are established analogously to the main axes. FIG. 1 showsthe basic quantities and the relative position of the flight frame andearth frame of references.

To simply describe the aerodynamic forces, an aerodynamic coordinatesystem F_(A) is selected, the origin of which is also located in thecentre of gravity C of the aircraft. The x-axis of this coordinatesystem lies in the direction of the negative oncoming flow speed and thez-axis is in the direction of the negative lift. The y-axis is selectedanalogously to the previous observations. This coordinate system isobtained by rotating the body frame main axis system by an angle ofattack a about its y-axis and then by an angle of yaw β about thez-axis. The aerodynamic coordinate system F_(A) is a body framereference only in stationary flying states of the aircraft.

The transition from the body frame to the earth frame of reference ismade using a transformation matrix L_(EB).

$\begin{matrix}{L_{EB} = \begin{bmatrix}{\cos\;\theta\;\cos\;\psi} & \begin{matrix}{{\sin\;\phi\;\sin\;{\theta cos}\;\psi} -} \\{\cos\;{\phi sin}\;\psi}\end{matrix} & {{\cos\;\phi\;\sin\;\phi\;\cos\;\psi} + {\sin\;{\phi cos}\;\psi}} \\{\cos\;\theta\;\sin\;\psi} & \begin{matrix}{{\sin\;\phi\;\sin\;\theta\;\sin\;\psi} +} \\{\cos\;\phi\;\cos\;\psi}\end{matrix} & {{\cos\;\phi\;\sin\;\theta\;\sin\;\psi} - {\sin\;\phi\;\cos\;\psi}} \\{{- \sin}\;\theta} & {\sin\;\phi\;\cos\;\theta} & {\cos\;{\phi cos}\;\theta}\end{bmatrix}} & (9)\end{matrix}$

The subscript index indicates the coordinate system in which the vectorsare presented. For example, the vector {right arrow over (R)}_(E) in theearth frame of reference F_(E) is obtained from the vector {right arrowover (R)}_(B) shown in body frame coordinates, where:{right arrow over (R)} _(E) =L _(EB) {right arrow over (R)} _(B)  (10)

To simplify the notation, the index B is omitted in the following if itis not absolutely necessary.

When considering speed, a distinction must also be made between wind andcalm. The following generally applies with the speed addition law:{right arrow over (V)} _(E) ^(E) ={right arrow over (V)} _(E) ^(B)+{right arrow over (W)} _(E)  (11)with the superscript index establishing the reference system in whichthe corresponding speeds are measured. {right arrow over (W)}_(E) is thewind speed which can be assumed as zero. Thus, the amounts in bothreference systems are the same and the superscript index can be omitted.

With the components of the vectors {right arrow over (V)}, {right arrowover (Ω)}, and {right arrow over (Φ)} as state quantities, the motionequations are obtained in state space during a calm from Newton'sequation and the angular momentum equation, as well as from therelationship between the Euler angles and the rates thereof. Theequations apply in particular when the earth is considered as aninertial system with a homogeneous gravitational field and the aeroplaneor aircraft is symmetrical with respect to its x-z plane. The arisingforces engage according to the model in the centre of gravity and thegeneration of aerodynamic forces is quasi-stationary.

Newton's equation for the centre of gravity of the aircraft in earthframe coordinates is:{right arrow over (F)} _(E) =m {right arrow over ({dot over (V)})}_(E)  (12)

Using the transformation matrix L _(EB), this is transformed into thebody frame of reference.

$\begin{matrix}\begin{matrix}{{{\underset{\_}{L}}_{EB}\overset{\rightarrow}{F}} = {m\frac{\mathbb{d}}{\mathbb{d}t}\left( {{\underset{\_}{L}}_{EB}\overset{\rightarrow}{V}} \right)}} \\{= {m\left( {{{\underset{\_}{\overset{.}{L}}}_{EB}\overset{\rightarrow}{V}} + {{\underset{\_}{L}}_{EB}\overset{\overset{.}{\rightarrow}}{V}}} \right)}}\end{matrix} & (13)\end{matrix}$

The following applies:{dot over (L)} _(EB) {right arrow over (V)}=L _(EB)({right arrow over(Ω)}×{right arrow over (V)})  (14)from which follows:L _(EB) {right arrow over (F)}=L _(EB) m({right arrow over (Ω)}×{rightarrow over (V)}+{right arrow over ({dot over (V)})})  (15)

The resulting force {right arrow over (F)} is composed of theaerodynamic force {right arrow over (R)} and the weight force {rightarrow over (G)}=L_(EB) ⁻¹{right arrow over (G)}_(E) . Theserelationships are inserted into the above equation and then resolvedaccording to {dot over (V)}.

$\begin{matrix}{\overset{\overset{.}{\rightarrow}}{V} = {{\frac{1}{m}\;\left( {\overset{\rightarrow}{R} + {{\underset{\_}{L}}_{EB}^{- 1}{\overset{\rightarrow}{G}}_{E}}} \right)} - {\overset{\rightarrow}{\Omega} \times \overset{\rightarrow}{V}}}} & (16)\end{matrix}$

Thus, the equations for the speeds are established. The relationshipsfor the rates are obtained analogously from the angular momentumequation with the angular momentum {right arrow over (H)} and theinertia sensor I.

$\begin{matrix}{{{\overset{\rightarrow}{Q}}_{E} = \overset{.}{{\overset{\rightarrow}{H}}_{E}}}\begin{matrix}{{{\underset{\_}{L}}_{EB}\overset{\rightarrow}{Q}} = {\frac{\mathbb{d}}{\mathbb{d}t}\left( {{\underset{\_}{L}}_{EB}{\overset{\rightarrow}{H}}_{E}} \right)}} \\{= {{{\overset{.}{\underset{\_}{L}}}_{EB}\overset{\rightarrow}{H}} + {{\underset{\_}{L}}_{EB}\overset{.}{\overset{\rightarrow}{H}}}}} \\{= {{\underset{\_}{L}}_{EB}\left( {{\overset{\rightarrow}{\Omega} \times \underset{\_}{I}\overset{\rightarrow}{\Omega}} + {\underset{\_}{I}\overset{\overset{.}{\rightarrow}}{\Omega}}} \right)}} \\{\overset{\overset{.}{\rightarrow}}{\Omega} = {{{\underset{\_}{I}}^{- 1}\overset{\rightarrow}{Q}} - {{\underset{\_}{I}}^{- 1}\overset{\rightarrow}{\Omega} \times \underset{\_}{I}\overset{\rightarrow}{\Omega}}}}\end{matrix}} & (17)\end{matrix}$

These relationships, divided up into components, produce together withthe equations between Euler angles and the rates thereof, the stateequations of a rigid aircraft.

$\begin{matrix}\begin{matrix}{\overset{.}{u} = {{\frac{1}{m}X} - {g\;\sin\;\theta} - {qw} + {r\;\upsilon}}} \\{\overset{.}{\upsilon} = {{\frac{1}{m}Y} + {g\;\cos\;{\theta sin}\;\phi} - {r\; u} + {p\; w}}} \\{\overset{.}{w} = {{\frac{1}{m}Z} + {g\;\cos\;{\theta cos}\;\phi} - {p\;\upsilon} + {qu}}} \\{\overset{.}{p} = {\frac{1}{{I_{z}I_{x}} - I_{zx}^{2}}\left\lbrack {{q\;{r\left( {{I_{y}I_{z}} - I_{z}^{2} - I_{zx}^{2}} \right)}} + {{qpI}_{zx}\left( {I_{z} + I_{x} - I_{y}} \right)} + {LI}_{z} + {NI}_{zx}} \right\rbrack}} \\{\overset{.}{q} = {\frac{1}{I_{y}}\left\lbrack {{{rp}\mspace{11mu}\left( {I_{z} - I_{x}} \right)} + {I_{zx}\left( {r^{2} - p^{2}} \right)} + M} \right\rbrack}} \\{{\overset{.}{r} = {\frac{1}{{I_{z}I_{x}} - I_{zx}^{2}}\left\lbrack {{{qrI}_{zx}\left( {I_{y} - I_{z} - I_{x}} \right)} + {{qp}\left( {I_{zx}^{2} + I_{x}^{2} - {I_{x}I_{y}}} \right)} + {LI}_{zx} + {NI}_{x}} \right\rbrack}}\;} \\{\overset{.}{\phi} = {p + {\left( {{q\;\sin\;\phi} + {r\;\cos\;\phi}} \right)\mspace{11mu}\tan\;\theta}}} \\{\overset{.}{\theta} = {{q\mspace{11mu}\cos\;\phi} - {r\;\sin\;\phi}}} \\{\overset{.}{\psi} = {\frac{1}{\cos\;\theta}\left( {{q\;\sin\;\phi} + {r\;\cos\;\phi}} \right)}}\end{matrix} & (17)\end{matrix}$

By transforming the speed {right arrow over (V)} into the earth frame ofreference, where{right arrow over (V)}=L _(EB) {right arrow over (V)}  (19)

the differential equations are obtained for calculating the position:{dot over (x)} _(E) =u cos θ cos ψ+υ(sin φ sin θ cos ψ−cos φ sinψ)+w(cos φ sin θ cos ψ+sin φ sin ψ){dot over (y)} _(E) =u cos θ sin ψ+υ(sin φ sin θ sin ψ+cos φ cosψ)+w(cos φ sin θ sin ψ−sin φ cos ψ)ż _(E) =−u sin θ+υ sin φ cos θ+w cos φ cos θ  (20)

For the specific force, the following is obtained in body framecoordinates for a sensor located on the x-axis at a spacing x_(p) fromthe centre of gravity:

$\begin{matrix}{\begin{bmatrix}b_{x} \\b_{y} \\b_{z}\end{bmatrix} = {\begin{bmatrix}\overset{.}{u} \\\overset{.}{\upsilon} \\\overset{.}{w}\end{bmatrix} - {g\begin{bmatrix}{{- \sin}\;\theta} \\{\sin\;\phi\mspace{14mu}\cos\;\theta} \\{\cos\;\phi\mspace{14mu}\cos\;\theta}\end{bmatrix}} + {x_{p}\begin{bmatrix}{- \left( {q^{2} + r^{2}} \right)} \\\overset{.}{r} \\{- \overset{.}{q}}\end{bmatrix}}}} & (21)\end{matrix}$

If the vector entries are divided by the gravitational acceleration

${g = {9.81\frac{m}{s}}},$the specific load factor is produced:n _(x) =b _(x) /g, n _(y) =b _(y) /g, n _(z) =b _(z) /g.

The above motion equations apply ideally to a rigid aircraft. However,in practice elastic deformations of the structure occur which have asignificant influence on the dynamic characteristics of the system.Therefore, the model is expanded by these elastic degrees of freedom.Quasi-static deformations are provided when the natural frequencies ofthe elastic modes are substantially higher than those of the rigid bodymodes. In this case, the influence of elastic deformation can beconsidered by a corresponding adaption of the aerodynamic derivatives.

If the natural frequencies of the elastic degrees of freedom are withinthe same range, the movement of the rigid body is influenced by theelastic deformations. In this case, the dynamics of the elastic degreesof freedom are to be considered in the motion equations. For thispurpose, the deformation of the structure can be approximately describedby the superposition of normal modes of the free vibration:

$\begin{matrix}{{{x^{\prime}(t)} = {\sum\limits_{n = 1}^{\infty}\;{{f_{n}\left( {x_{0},y_{0},z_{0}} \right)}{ɛ_{n}(t)}}}}{{y^{\prime}(t)} = {\sum\limits_{n = 1}^{\infty}\;{{g_{n}\left( {x_{0},y_{0},z_{0}} \right)}{ɛ_{n}(t)}}}}{{z^{\prime}(t)} = {\sum\limits_{n = 1}^{\infty}\;{{h_{n}\left( {x_{0},y_{0},z_{0}} \right)}{ɛ_{n}(t)}}}}} & (22)\end{matrix}$

x′, y′ z′ are the deflections of the respective rest positions x_(o),y_(o), z_(o); f_(n), g_(n) and h_(n) are the mode form functions andε_(n) are generalised coordinates. The additional motion equations formode ε_(n) are obtained from the Lagrange equation as equations offorced vibrations. For the mode ε_(n), the following approximatelyapplies with the natural frequency ω_(n) of the damping d_(n) and thegeneralised moment of inertia I_(n)

$\begin{matrix}{{{\overset{¨}{ɛ}}_{n} + {2d_{n}\omega_{n}\overset{.}{ɛ}} + {\omega_{n}^{2}ɛ_{n}}} = \frac{F_{n}}{I_{n}}} & (23)\end{matrix}$

The approximation consists in disregarding all couplings over thedamping terms between the individual modes. On the assumption that theinfluence of the degrees of freedom of the rigid body on the elasticmodes can be described by a linear correlation and that the elasticdeformations are adequately small, the generalised force F_(n) ispresented as a linear combination of state and input quantities:

$\begin{matrix}{F_{n} = {{a_{nu}\Delta\; u} + {a_{n\;\overset{.}{u}}\overset{.}{u}} + \ldots + {a_{np}p} + \ldots + {a_{n\;\delta_{r}}\delta_{r}} + \ldots + {\sum\limits_{j = 1}^{\infty}\;{a_{nj}ɛ_{j}}} + {\sum\limits_{j = 1}^{\infty}\;{b_{nj}{\overset{.}{ɛ}}_{j}}} + {\sum\limits_{j = 1}^{\infty}\;{c_{nj}{\overset{¨}{ɛ}}_{j}}}}} & (24)\end{matrix}$

The infinite series occurring here can be replaced by finite serieswhich only retain those modes which lie in the range of the rigid bodyfrequencies. It can be assumed for the further calculation that theseare K modes which are combined in a vector ε. Therefore, equation (24)can be written in the following form:

$\begin{matrix}\begin{matrix}{F_{n} = {{a_{nu}\Delta\; u} + {a_{n\overset{.}{u}}\overset{.}{u}} + \ldots + {a_{np}p} + \ldots + {a_{n\;\delta_{r}}\delta_{r}} + \ldots +}} \\{{\sum\limits_{j = 1}^{k}\;{a_{nj}ɛ_{j}}} + {\sum\limits_{j = 1}^{k}{b_{nj}{\overset{.}{ɛ}}_{j}}} + {\sum\limits_{j = 1}^{k}\;{c_{nj}{\overset{¨}{ɛ}}_{j}}}} \\{= {{a_{nu}\Delta\; u} + {a_{n\overset{.}{u}}\overset{.}{u}} + \ldots + {a_{np}p} + \ldots + {a_{n\;\delta_{r}}\delta_{r}} + \ldots +}} \\{{{\underset{\_}{a}}_{n\; ɛ}^{T}\underset{\_}{ɛ}} + {{\underset{\_}{b}}_{n\;\overset{.}{ɛ}}^{T}\overset{.}{\underset{\_}{ɛ}}} + {{\underset{\_}{c}}_{n\;\overset{¨}{ɛ}}^{T}\underset{\_}{\overset{¨}{ɛ}}}}\end{matrix} & (25)\end{matrix}$

To arrive at a compact notation for all modes, the generalised momentsof inertia I_(n) are combined into the diagonal matrix I, the scalarcouplings are each combined into vectors and the vectorial couplingterms are combined into matrices. Thus, equation (24) can be formulatedfor all modes.{umlaut over (ε)}+2 d ω ^(T){dot over (ε)}+ω ω ^(T) ε=I ⁻¹( a _(u) Δu+a_({dot over (u)}) {dot over (u)}+ . . . +a _(p) p+ . . . +a _(δ) _(r)δ_(r) + . . . +A _(ε) ε+B _({dot over (ε)}) {dot over (ε)}+C _(ε){umlaut over (ε)})  (26)

Presentation in the state space is achieved by introducing the modespeed {dot over (ε)}=υ. This is used in equation (26){dot over (υ)}+2 d ω ^(T) υ+ω ω ^(T) ε=I ⁻¹( a _(u) Δu+a_({dot over (u)}) {dot over (u)}+ . . . +a _(p) p+ . . . +a _(δ) _(r)δ_(r)+ . . . . + A _(ε) ε+ B _(υ) υ+ C _({dot over (υ)}){dot over(υ)})  (27)

Using the MatricesA _({dot over (x)}) ₁ =[a _({dot over (u)}) a _({dot over (υ)}) a_({dot over (w)}) a _({dot over (p)}) a _({dot over (q)}) a_({dot over (r)})],A _(x) ₁ =[a _(u) a _(υ) a _(w) a _(p) a _(q) a _(r)],A _(c)=[a _(δ) _(E) a _(δ) _(A) a _(δ) _(R) a _(δ) _(c) a _(δ) _(F)],  (28)and the unit matrix of k order I_(k) , the state equation can beformulated:{dot over (ε)}=υ{dot over (υ)}=( I _(k) −I ⁻¹ C _({dot over (υ)}))⁻¹[( I ⁻¹ B _(υ)−2 d ω^(T))υ+( I ⁻¹ A _(ε)−ω ω ^(T))ε+ A _({dot over (x)}) ₁ x ₁ +A _(x) ₁ x ₁+A _(c) c]  (29)

The external forces acting on an aircraft are, in addition to theweight, the aerodynamic forces of lift and resistance as well as thrust.The point of application of the lift is in what is known as the neutralpoint which is different from the centre of gravity. As a result,moments are generated. This applies similarly to thrust. The resultingforces are combined in a vector {right arrow over (R)} and the momentsare combined in a vector {right arrow over (Q)}. Lift and resistance aregenerated by the relative movement of aircraft and air, i.e. by {rightarrow over (V)} and {right arrow over (Ω)}. Furthermore, these forcesdepend on the angle of attack α and the angles of the control surfacesof the primary flight control, elevator (δ_(E)), aileron (δ_(A)) andrudder (δ_(R)). Depending on the type of aircraft, further controlsurfaces, spoilers, canards are used which are denoted in the followingby δ_(c). The angles of the control surfaces are combined together withthe thrust δ_(F) in a control, vector c. The aerodynamic effects arebased on non-linear correlations. They can be described by Taylor'sseries which are interrupted according to a specific order. Thecoefficients of members of the second and third order are below thefirst order coefficients by one to two orders of magnitude. If the angleof attack remains below 10°, the terms of a higher order can bedisregarded. The starting point for the linear approach is a stationaryflight state. The speeds and rates as well as forces and moments aresplit into a stationary and a disturbance term:u=u ₀ +Δu X=X ₀ +ΔX p=p ₀ +Δp L=L ₀ +ΔLυ=υ₀ +ΔυY=Y ₀ +ΔY q=q ₀ +Δq M=M ₀ +ΔMw=w ₀ +Δw Z=Z ₀ +ΔZ r=r ₀ +Δr N=N ₀ +ΔN  (30)

The horizontal symmetrical straight flight can be selected as thestationary flight state. If the stability axes are additionally selectedas a flight frame of reference, the above relationships are simplifiedin that in this state, X₀=Y₀=L₀=M₀=N₀=0 und ω₀=u₀=μ₀=q₀=r₀=0. Since inhorizontal flight, the z-axes of the flight frame and earth frame ofreference are parallel, then Z₀=−mg. Furthermore, it approximatelyapplies that ω≈u₀α.

$\begin{matrix}{\begin{bmatrix}X \\Z \\M \\Y \\L \\N\end{bmatrix} = {\begin{bmatrix}0 \\{{- m}\; g} \\0 \\0 \\0 \\0\end{bmatrix} + {\quad{\left\lbrack \begin{matrix}X_{u} & X_{w} & X_{\overset{.}{w}} & X_{q} & 0 & 0 & 0 & 0 & {\underset{\_}{X}}_{ɛ} & {\underset{\_}{X}}_{v} & {\underset{\_}{X}}_{c} \\Z_{u} & Z_{w} & Z_{\overset{.}{w}} & Z_{q} & 0 & 0 & 0 & 0 & {\underset{\_}{Z}}_{ɛ} & {\underset{\_}{Z}}_{v} & {\underset{\_}{Z}}_{c} \\M_{u} & M_{w} & M_{\overset{.}{w}} & M_{q} & 0 & 0 & 0 & 0 & {\underset{\_}{M}}_{ɛ} & {\underset{\_}{M}}_{v} & {\underset{\_}{M}}_{c} \\0 & 0 & 0 & 0 & Y_{\upsilon} & Y_{\overset{.}{\upsilon}} & Y_{p} & Y_{r} & {\underset{\_}{Y}}_{ɛ} & {\underset{\_}{Y}}_{v} & {\underset{\_}{Y}}_{c} \\0 & 0 & 0 & 0 & L_{\upsilon} & L_{\overset{.}{\upsilon}} & L_{p} & L_{r} & {\underset{\_}{L}}_{ɛ} & {\underset{\_}{L}}_{v} & {\underset{\_}{L}}_{c} \\0 & 0 & 0 & 0 & N_{\upsilon} & N_{\overset{.}{\upsilon}} & N_{p} & N_{r} & {\underset{\_}{N}}_{ɛ} & {\underset{\_}{N}}_{v} & {\underset{\_}{N}}_{c}\end{matrix} \right\rbrack\begin{bmatrix}{\Delta\; u} \\w \\\overset{.}{w} \\q \\\upsilon \\\overset{.}{\upsilon} \\p \\r \\\underset{\_}{ɛ} \\\underset{\_}{v} \\\underset{\_}{c}\end{bmatrix}}}}} & (31)\end{matrix}$

The quantities which occur in equation (31) and are indexed with u and εdescribe the influence of the elastic modes on the aerodynamics. Theyare in each case vectors of length k, with k being the number of elasticmodes. The derivatives indexed with c are also vectors which describethe influence of the control factors. The dimension thereof is equal tothe number of control factors.

The equations derived above are combined into a model by which theentire dynamics of the flexible aircraft can be described under theconditions mentioned in the preceding paragraphs. The states fordescribing the movement of the rigid body are combined in the vectorx ₁ =[Δu w q θ υ p r φ ψ] ^(T)  (32)

ε and υ denote the introduced elastic modes, while the control factorsare contained in the vector c. As for the introduction of theaerodynamic forces, this case also originates from the symmetricalhorizontal straight flight. All the disturbance terms are assumed to besmall enough for the linear approximation to be valid for theaerodynamics. Furthermore, A _({dot over (x)}1) is disregarded. Underthese conditions, the motion equations can be written in the followingform:

$\begin{matrix}{\begin{bmatrix}{\underset{\_}{\overset{.}{x}}}_{1} \\\overset{.}{\underset{\_}{ɛ}} \\\underset{\_}{\overset{.}{v}}\end{bmatrix} = {{\begin{bmatrix}{\underset{\_}{A}}_{11} & {\underset{\_}{A}}_{12} & {\underset{\_}{A}}_{13} \\\underset{\_}{0} & \underset{\_}{0} & {\underset{\_}{I}}_{k} \\{\underset{\_}{A}}_{31} & {\underset{\_}{A}}_{32} & {\underset{\_}{A}}_{33}\end{bmatrix}\begin{bmatrix}{\underset{\_}{x}}_{1} \\\underset{\_}{ɛ} \\\underset{\_}{v}\end{bmatrix}} + {\begin{bmatrix}{\underset{\_}{B}}_{1} \\0 \\{\underset{\_}{B}}_{3}\end{bmatrix}\underset{\_}{c}} + {\begin{bmatrix}\underset{\_}{F} \\\underset{\_}{0} \\\underset{\_}{0}\end{bmatrix}{\underset{\_}{g}\left( {\underset{\_}{x}}_{1} \right)}}}} & (33) \\{b = {{\begin{bmatrix}{\underset{\_}{C}}_{1} & {\underset{\_}{C}}_{2} & {\underset{\_}{C}}_{3}\end{bmatrix}\begin{bmatrix}{\underset{\_}{x}}_{1} \\\underset{\_}{ɛ} \\\underset{\_}{v}\end{bmatrix}} + {\underset{\_}{H}\;{\underset{\_}{h}\left( {\underset{\_}{x}}_{1} \right)}} + {\underset{\_}{D}\;\underset{\_}{c}}}} & (34)\end{matrix}$

The partial matrices used in equations (33) and (34) are compiled withthe following abbreviations:

$\begin{matrix}{{\Delta = {{I_{z}I_{x}} - I_{zx}^{2}}}{I_{{qr}\; 1} = {{I_{y}I_{z}} - I_{z}^{2} - I_{zx}^{2}}}{I_{{pq}\; 1} = {I_{zx}\left( {I_{z} + I_{x} - I_{y}} \right)}}{I_{{qr}\; 2} = {I_{zx}\left( {I_{y} - I_{z} - I_{x}} \right)}}{I_{{pq}\; 2} = {I_{zx}^{2} + I_{x}^{2} - {I_{x}I_{y}}}}{m_{\overset{.}{w}} - m - Z_{\overset{.}{w}}}{m_{\overset{.}{\upsilon}} = {m - Y_{\overset{.}{\upsilon}}}}{L_{i}^{\prime} = {{I_{z}L_{i}} + {I_{zx}N_{i}}}}{N_{i}^{\prime} = {{I_{zx}L_{i}} + {I_{x}N_{i}}}}} & (35) \\{{\underset{\_}{A}}_{11} = \begin{bmatrix}{\underset{\_}{A}}_{long} & 0 \\0 & {\underset{\_}{A}}_{lat}\end{bmatrix}} & (36) \\{{\underset{\_}{A}}_{long} = {\quad\left\lbrack \begin{matrix}{\frac{X_{u}}{m} + \frac{Z_{u}X_{\overset{.}{w}}}{m_{\overset{.}{w}}m}} & {\frac{X_{w}}{m} + \frac{Z_{w}X_{\overset{.}{w}}}{m_{\overset{.}{w}}m}} & {\frac{X_{q}}{m} + \frac{X_{\overset{.}{w}}\left( {Z_{q} + {mu}_{0}} \right)}{m_{\overset{.}{w}}m}} & 0 \\\frac{Z_{u}}{m_{\overset{.}{\omega}}} & \frac{Z_{w}}{m_{\overset{.}{w}}} & \frac{Z_{q} + {m\; u_{0}}}{m_{\overset{.}{w}}} & 0 \\{\frac{1}{I_{y}}\left( {M_{u} + \frac{M_{\overset{.}{w}}Z_{u}}{m_{\overset{.}{w}}}} \right)} & {\frac{1}{I_{y}}\left( {M_{w} + \frac{M_{\overset{.}{w}}Z_{w}}{m_{\overset{.}{w}}}} \right)} & {\frac{1}{I_{y}}\left( {M_{q} + \frac{M_{\overset{.}{w}}\left( {Z_{q} + {m\; u_{0}}} \right)}{m_{\overset{.}{w}}}} \right)} & 0 \\0 & 0 & 0 & 0\end{matrix} \right\rbrack}} & (37) \\{{\underset{\_}{A}}_{lat} = \begin{bmatrix}\frac{Y_{\upsilon}}{m_{\overset{.}{\upsilon}}} & \frac{Y_{p}}{m_{\overset{.}{\upsilon}}} & \frac{Y_{r} - {m\; u_{0}}}{m_{\overset{.}{\upsilon}}} & 0 & 0 \\{\frac{L_{\upsilon}^{\prime}}{\Delta} + {Y_{\upsilon}\frac{L_{\overset{.}{\upsilon}}^{\prime}}{\Delta\; m_{\overset{.}{\upsilon}}}}} & {\frac{L_{p}^{\prime}}{\Delta} + {Y_{p}\frac{L_{\overset{.}{\upsilon}}^{\prime}}{\Delta\; m_{\overset{.}{\upsilon}}}}} & {\frac{L_{r}^{\prime}}{\Delta} + \frac{L_{\overset{.}{\upsilon}}^{\prime}\left( {Y_{r} - {m\; u_{0}}} \right)}{\Delta\; m_{\overset{.}{\upsilon}}}} & 0 & 0 \\{\frac{N_{\upsilon}^{\prime}}{\Delta} + {Y_{\upsilon}\frac{N_{\overset{.}{\upsilon}}^{\prime}}{\Delta\; m_{\overset{.}{\upsilon}}}}} & {\frac{N_{p}^{\prime}}{\Delta} + {Y_{p}\frac{N_{\overset{.}{\upsilon}}^{\prime}}{\Delta\; m_{\overset{.}{\upsilon}}}}} & {\frac{N_{r}^{\prime}}{\Delta} + \frac{N_{\overset{.}{\upsilon}}^{\prime}\left( {Y_{r} - {m\; u_{0}}} \right)}{\Delta\; m_{\overset{.}{\upsilon}}}} & 0 & 0 \\0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0\end{bmatrix}} & (38) \\{{{\underset{\_}{A}}_{12} = \begin{bmatrix}{\underset{\_}{A}}_{12\;{long}} & {\underset{\_}{A}}_{12{lat}}\end{bmatrix}^{T}}{where}} & (39) \\{{{\underset{\_}{A}}_{12\;{long}} = \begin{bmatrix}{{\frac{{\underset{\_}{X}}_{ɛ}^{T}}{m} + \frac{{\underset{\_}{Z}}_{ɛ}X_{\overset{.}{w}}}{m_{\overset{.}{w}}m}}\mspace{11mu}} & {\frac{{\underset{\_}{Z}}_{ɛ}^{T}}{m_{\overset{.}{w}}}\;} & \frac{1}{I_{y}} & \left( {{\underset{\_}{M}}_{ɛ} + \frac{M_{\overset{.}{w}}{\underset{\_}{Z}}_{ɛ}}{m_{\overset{.}{w}}}} \right)^{T} & {\underset{\_}{0}}^{T}\end{bmatrix}}{{\underset{\_}{A}}_{12\;{lat}} = \begin{bmatrix}\frac{{\underset{\_}{Y}}_{ɛ}^{T}}{m_{\overset{.}{\upsilon}}} & \frac{1}{\Delta} & \left( {{\underset{\_}{L}}_{ɛ}^{\prime} + {{\underset{\_}{Y}}_{ɛ}\frac{L_{\overset{.}{\upsilon}}^{\prime}}{\Delta\; m_{\overset{.}{\upsilon}}}}} \right)^{T} & \frac{1}{\Delta} & \left( {{\underset{\_}{N}}_{ɛ}^{\prime} + {{\underset{\_}{Y}}_{ɛ}\frac{N_{\overset{.}{\upsilon}}^{\prime}}{\Delta\; m_{\overset{.}{\upsilon}}}}} \right)^{T} & {\underset{\_}{0}}^{T} & {\underset{\_}{0}}^{T}\end{bmatrix}}} & (40)\end{matrix}$

The matrices A₁₃ and B₁ are obtained by respectively replacing the indexε in the matrix A₁₂ by u and c. The following applies to the remainingmatrices:

$\begin{matrix}{{{\underset{\_}{A}}_{3l} = {\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)^{- 1}{\underset{\_}{A}}_{x_{1}}}},{{\underset{\_}{A}}_{32} = {\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)^{- 1}\left( {{{\underset{\_}{I}}^{- 1}{\underset{\_}{A}}_{ɛ}} - {\underset{\_}{\omega}\mspace{11mu}\underset{\_}{\omega^{T}}}} \right)}},{{\underset{\_}{A}}_{33} = {\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)^{- 1}\left( {{{\underset{\_}{I}}^{- 1}{\underset{\_}{B}}_{\upsilon}} - {2\underset{\_}{d}\;\underset{\_}{\omega^{T}}}} \right)}},{{\underset{\_}{B}}_{3} = {\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)^{- 1}{\underset{\_}{A}}_{c}}},} & (41) \\{F = \begin{bmatrix}{- g} & \frac{{gX}_{\overset{.}{w}}}{m_{w}} & 0 & 0 & 0 & 0 & 1 & \frac{X_{\overset{.}{w}}}{m_{w}} & 0 & 0 & 0 & 0 & 0 \\0 & \frac{m\; g}{m_{\overset{.}{w}}} & 0 & 0 & 0 & 0 & 0 & \frac{m}{m_{\overset{.}{w}}} & 0 & 0 & 0 & 0 & 0 \\0 & \frac{M_{\overset{.}{w}}m\; g}{I_{y}m_{\overset{.}{w}}} & 0 & 0 & 0 & 0 & 0 & \frac{M_{\overset{.}{w}}m}{I_{y}m_{\overset{.}{w}}} & 0 & 0 & 0 & \frac{I_{z} - I_{x}}{I_{y}} & \frac{I_{zx}}{I_{y}} \\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & \frac{m\; g}{m_{\overset{.}{v}}} & 0 & 0 & 0 & 0 & \frac{m}{m_{\overset{.}{v}}} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & \frac{{mgL}_{\overset{.}{v}}^{\prime}}{\Delta\; m_{\overset{.}{v}}} & 0 & 0 & 0 & 0 & \frac{m\; L_{\overset{.}{v}}}{\Delta\; m_{\overset{.}{v}}} & \frac{I_{{qr}\; 1}}{\Delta} & \frac{I_{{pq}\; 1}}{\Delta} & 0 & 0 \\0 & 0 & 0 & \frac{{mgN}_{\overset{.}{v}}^{\prime}}{\Delta\; m_{\overset{.}{v}}} & 0 & 0 & 0 & 0 & \frac{m\; N_{\overset{.}{v}}}{\Delta\; m_{\overset{.}{v}}} & \frac{I_{{qr}\; 2}}{\Delta} & \frac{I_{{pq}\; 2}}{\Delta} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix}} & (42) \\{{\underset{\_}{g}\left( {\underset{\_}{x}}_{1} \right)} = \begin{bmatrix}{\sin\;\theta} \\{{\cos\;{\theta cos}\;\phi} - 1} \\{{q\;\cos\;\phi} - {r\;\sin\;\phi}} \\{\cos\;{\theta sin}\;\phi} \\{\left( {{q\;\sin\;\phi} + {r\;\cos\;\phi}} \right)\tan\;\theta} \\{\frac{1}{\cos\;\theta}\left( {{q\;\sin\;\phi} + {r\;\cos\;\phi}} \right)} \\{{- {qw}} + {r\;\upsilon}} \\{{{- p}\;\upsilon} + {q\;\Delta\; u}} \\{{{- r}\;\Delta\; u} + {p\; w}} \\{qr} \\{pq} \\{rp} \\{r^{2} - p^{2}}\end{bmatrix}} & (43) \\{{{\underset{\_}{C}}_{1} = \left\lfloor {{\underset{\_}{C}}_{1\;{long}}\mspace{14mu}{\underset{\_}{C}}_{1\;{lat}}} \right\rfloor},} & (44) \\{{{\underset{\_}{C}}_{1\;{long}} = \left\lbrack \begin{matrix}\begin{matrix}{\frac{X_{u}}{m} + \frac{X_{\overset{.}{w}}Z_{u}}{m\; m_{\overset{.}{w}}} +} \\{{{\underset{\_}{K}}_{x}\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{u}}\end{matrix} & \; & \begin{matrix}{\frac{X_{w}}{m} + \frac{X_{\overset{.}{w}}Z_{w}}{m\; m_{\overset{.}{w}}} +} \\{{{\underset{\_}{K}}_{x}\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{w}}\end{matrix} & \begin{matrix}{\frac{X_{q}}{m} + \frac{X_{\overset{.}{w}}\left( {Z_{q} + {m\; u_{0}}} \right)}{m\; m_{\overset{.}{w}}} +} \\{{{\underset{\_}{K}}_{x}\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{q}}\end{matrix} & 0 \\\begin{matrix}{\frac{Z_{u}}{m_{\overset{.}{w}}} - {\frac{x_{p}}{I_{y}}\left( {M_{u} + \frac{M_{\overset{.}{w}}Z_{u}}{m_{\overset{.}{w}}}} \right)} +} \\{{{\underset{\_}{K}}_{z}\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{u}}\end{matrix} & \; & \begin{matrix}{\frac{Z_{w}}{m_{\overset{.}{w}}} - {\frac{x_{p}}{I_{y}}\left( {M_{w} + \frac{M_{\overset{.}{w}}Z_{w}}{m_{\overset{.}{w}}}} \right)} +} \\{{{\underset{\_}{K}}_{z}\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{w}}\end{matrix} & \begin{matrix}{\frac{Z_{q} + {m\; u_{0}}}{m_{\overset{.}{w}}} -} \\{{\frac{x_{p}}{I_{y}}\left( {M_{q} + \frac{M_{\overset{.}{w}}\left( {Z_{q} + {m\; u_{0}}} \right)}{m_{\overset{.}{w}}}} \right)} +} \\{{{\underset{\_}{K}}_{z}\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{q}}\end{matrix} & 0 \\{{{\underset{\_}{K}}_{y}\left( {{\underset{\_}{I}}_{k}^{- 1} - {\underset{\_}{I}\;{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{u}} & \; & {{{\underset{\_}{K}}_{y}\left( {{\underset{\_}{I}}_{k}^{- 1} - {\underset{\_}{I}\mspace{11mu}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{w}} & {{{\underset{\_}{K}}_{y}\left( {{\underset{\_}{I}}_{k}^{- 1} - {\underset{\_}{I}\mspace{11mu}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{q}} & 0\end{matrix} \right\rbrack},} & (45) \\{{{\underset{\_}{C}}_{1\;{lat}} = \left\lbrack \begin{matrix}{{{\underset{\_}{K}}_{x}\left( {{\underset{\_}{I}}_{k}^{- 1} - {\underset{\_}{I}\;{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{v}} & \; & \; & {{{\underset{\_}{K}}_{x}\left( {{\underset{\_}{I}}_{k}^{- 1} - {\underset{\_}{I}\;{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{p}} & {{{\underset{\_}{K}}_{x}\left( {{\underset{\_}{I}}_{k}^{- 1} - {\underset{\_}{I}\;{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{y}} & 0 & 0 \\{{{\underset{\_}{K}}_{z}\left( {{\underset{\_}{I}}_{k}^{- 1} - {\underset{\_}{I}\;{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{v}} & \; & \; & {{{\underset{\_}{K}}_{z}\left( {{\underset{\_}{I}}_{k}^{- 1} - {\underset{\_}{I}\;{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{p}} & {{{\underset{\_}{K}}_{z}\left( {{\underset{\_}{I}}_{k}^{- 1} - {\underset{\_}{I}\;{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{y}} & 0 & 0 \\\begin{matrix}{\frac{Y_{v}}{m_{\overset{.}{v}}} + {\frac{x_{p}}{\Delta}\left( {N_{v}^{\prime} + \frac{N_{\overset{.}{v}}^{\prime}Y_{v}}{m_{\overset{.}{v}}}} \right)} +} \\{{{\underset{\_}{K}}_{y}\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{v}}\end{matrix} & \; & \; & \begin{matrix}{\frac{Y_{p}}{m_{\overset{.}{v}}} + {\frac{x_{p}}{\Delta}\left( {N_{p}^{\prime} + \frac{N_{\overset{.}{v}}^{\prime}Y_{p}}{m_{\overset{.}{v}}}} \right)} +} \\{{{\underset{\_}{K}}_{y}\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{p}}\end{matrix} & \begin{matrix}{\frac{Y_{y} - {m\; u_{0}}}{m_{\overset{.}{v}}} +} \\{{\frac{x_{p}}{\Delta}\left( {N_{r}^{\prime} + \frac{N_{\overset{.}{v}}^{\prime}\left( {Y_{y} - {m\; u_{0}}} \right)}{m_{\overset{.}{v}}}} \right)} +} \\{{{\underset{\_}{K}}_{y}\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}{\underset{\_}{a}}_{y}}\end{matrix} & 0 & 0\end{matrix} \right\rbrack},} & (46) \\{{{\underset{\_}{C}}_{2} = \begin{bmatrix}{\frac{{\underset{\_}{X}}_{ɛ}}{m} + \frac{X_{\overset{.}{w}}{\underset{\_}{Z}}_{ɛ}}{m\; m_{\overset{.}{w}}} + {{{\underset{\_}{K}}_{x}\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}\left( {{{\underset{\_}{I}}^{- 1}{\underset{\_}{A}}_{ɛ}} - {\underset{\_}{\omega}\mspace{11mu}{\underset{\_}{\omega}}^{T}}} \right)}} \\{\frac{{\underset{\_}{Z}}_{ɛ}}{m_{\overset{.}{w}}} - {\frac{x_{p}}{I_{y}}\left( {{\underset{\_}{M}}_{ɛ} + \frac{M_{\overset{.}{w}}{\underset{\_}{Z}}_{ɛ}}{m_{\overset{.}{w}}}} \right)} + {{{\underset{\_}{K}}_{z}\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}\left( {{{\underset{\_}{I}}^{- 1}{\underset{\_}{A}}_{ɛ}} - {\underset{\_}{\omega}\mspace{11mu}{\underset{\_}{\omega}}^{T}}} \right)}} \\{\frac{{\underset{\_}{Y}}_{ɛ}}{m_{\overset{.}{\upsilon}}} - {\frac{x_{p}}{\Delta}\left( {{\underset{\_}{N}}_{ɛ}^{\prime} + \frac{N_{\overset{.}{\upsilon}}^{\prime}{\underset{\_}{Y}}_{ɛ}}{m_{\overset{.}{\upsilon}}}} \right)} + {{{\underset{\_}{K}}_{y}\left( {{\underset{\_}{I}}_{k} - {{\underset{\_}{I}}^{- 1}{\underset{\_}{C}}_{\overset{.}{v}}}} \right)}^{- 1}\left( {{{\underset{\_}{I}}^{- 1}{\underset{\_}{A}}_{ɛ}} - {\underset{\_}{\omega}\mspace{11mu}{\underset{\_}{\omega}}^{T}}} \right)}}\end{bmatrix}},} & (47)\end{matrix}$

The matrices C₃ and D are obtained by replacing the index ε by u and c,respectively. H and h(x₁) are as follows:

$\begin{matrix}{{\underset{\_}{H} = \left\lbrack \begin{matrix}\frac{X_{\overset{.}{w}}g}{m_{\overset{.}{w}}} & 0 & 0 & 1 & {- \frac{X_{\overset{.}{w}}}{m_{\overset{.}{w}}}} & 0 & 0 & 0 & 0 & {- x_{p}} & {- x_{p}} \\{\frac{m\; g}{m_{\overset{.}{w}}}\begin{pmatrix}{1 -} \\\frac{x_{p}M_{\overset{.}{w}}}{I_{y}}\end{pmatrix}} & {- g} & 0 & 0 & {\frac{m}{m_{\overset{.}{w}}}\begin{pmatrix}{1 -} \\\frac{x_{p}M_{\overset{.}{w}}}{I_{y}}\end{pmatrix}} & 0 & 0 & 0 & {{- \frac{x_{p}}{I_{y}}}\begin{pmatrix}{I_{z} -} \\I_{x}\end{pmatrix}} & {- \frac{x_{p}I_{zx}}{I_{y}}} & \frac{x_{p}I_{zx}}{I_{y}} \\0 & 0 & \begin{pmatrix}{\frac{Y_{\overset{.}{\upsilon}}g}{m_{\overset{.}{\upsilon}}} +} \\\frac{x_{p}N_{\overset{.}{\upsilon}}^{\prime}m\; g}{\Delta\; m_{\overset{.}{\upsilon}}}\end{pmatrix} & 0 & 0 & {\frac{m}{m_{\overset{.}{\upsilon}}}\begin{pmatrix}{1 +} \\\frac{x_{p}N_{\overset{.}{\upsilon}}^{\prime}}{\Delta}\end{pmatrix}} & \frac{x_{p}I_{{qr}\; 2}}{\Delta} & \frac{x_{p}I_{{pq}\; 2}}{\Delta} & 0 & 0 & 0\end{matrix} \right\rbrack},} & (48) \\{{\underset{\_}{h}\left( {\underset{\_}{x}}_{1} \right)} = \begin{bmatrix}{{\cos\;\theta\mspace{11mu}\cos\;\phi} - 1} \\{\cos\;\theta\mspace{11mu}\cos\;\phi} \\{\cos\;\theta\mspace{11mu}\sin\;\phi} \\{{{- q}\; w} + {r\;\upsilon}} \\{{{- p}\;\upsilon} + {q\;\Delta\; u}} \\{{{- r}\;\Delta\; u} + {p\; w}} \\{qr} \\{pq} \\{rp} \\r^{2} \\p^{2}\end{bmatrix}} & (49)\end{matrix}$

The non-linear simulation model described in equation (33) contains aneffectiveness matrix F which considers the non-linear characteristics ofparameters. The effectiveness matrix F is stated in equation (42).

Expanding the model by aerodynamic, structural dynamic and aeroelasticnon-linearities produces

a.) additional entries in the non-linearity vector g(x1),for exampleg₁₄(w)=w²+w¹⁴, g₁₅(v)=v², g₁₆(v₁)=v₁ ², g₁₇(v₂)=sgn (v₂), where sgn isthe so-called signum function of mathematics, and

b.) additional columns in the

$\quad\begin{bmatrix}\underset{\_}{F} \\\underset{\_}{0} \\\underset{\_}{0}\end{bmatrix}$matrix from equation (33):

The quantities X_(NL,w), Z_(NL,w), Y_(NL,w), D_(NL,1) and D_(NL,2)describe the influence strength of the non-linearity.

The non-linear simulation model presented in equation (33) can also bedescribed in a physically more concrete manner (in a generalisation ofthe Newton and Euler motion equations) as follows:M{umlaut over (x)}+D{dot over (x)}+Kx+Fg(x,{dot over (x)},p,t)=p  (50)wherex=[x_flight mechanics, x_system,x_aeroelastics]p=[p_gust, p_pilot_engine, p_fault]Fg(x,{dot over (x)},p,t) contains all non-linearities from flightmechanics, aerodynamics, systems, engine,  (51)and whereM: Expanded mass matrixD: Expanded damping matrixK: Expanded rigidity matrix.

Since the transformation of equation (33) into the form of equation (50)results in modified vectors x and g (x, {dot over (x)}, p, t) and amodified matrix F, these new vectors and matrices are not underlined.

The equation system is illustrated graphically in the diagram of FIG.4A. The equation system shown in FIG. 4A comprises a dynamic model oflinear differential equations which is expanded by an effectivenessmatrix F which is multiplied by a non-linearity vector g.

Positioned on the right-hand side of the equation system is ahyper-input vector p of the aircraft with a plurality of subvectors. Thevector x forms a hyper-momentum vector of the aircraft. The second timederivative x^(••) of the hyper movement vector x multiplied by anexpanded mass matrix M plus the first time derivative x^(•) of the hypermovement vector x multiplied by an expanded damping matrix D plus theproduct of a rigidity matrix K and the hyper movement vector x plus theproduct of the effectiveness matrix F with the non-linearity vector gproduces the hyper-input vector p of the aircraft.

Further non-linear expansions can be presented very graphically in thisillustration. Additional non-linearities in the engine dynamics, in thesystem behavior or in the case of faults expand the non-linearity vectorg (x, {dot over (x)}, p, t) and the effectiveness matrix F in additionalentries. The matrix entries of F in turn describe the influence strengthof non-linearities, as an effective force or moment in the generalisedNewton and Euler motion equations.

The mass matrix M, the damping matrix D and the rigidity matrix K areexpanded matrices which consider the aerodynamics.

FIG. 4B illustrates the structure of such an expanded matrix. Thecoupling describes the influence strength of a characteristic quantityon the aircraft. The mass matrix M″, the damping matrix D and therigidity matrix K describe linear influences, while the effectivenessmatrix F describes non-linear characteristics of parameters. Thesecharacteristic quantities are flight-mechanical characteristics,characteristic quantities of the on-board system and characteristicquantities of the aeroelastics.

FIGS. 5A, 5B, 5C, 5D and 5E show special cases of the non-linearsimulation model shown in a general manner in FIG. 4A. In the specialcase shown in FIG. 5A, the non-linear effectiveness matrix and thenon-linearity vector g and the input quantity vector p are zero. In thisway, the special case is reached of the purely linear equation system ofdifferential equations.

In the special case shown in FIG. 5B, the non-linear effectivenessmatrix and the non-linearity vector g are zero, while the input quantityvector p is not zero, for example to illustrate a gust of wind. Thus,the simulation model shown in FIG. 5B is suitable, for example, for theanalysis of gusts of wind which act on the aircraft.

In the special case shown in FIG. 5C, only flight-mechanical quantitiesare considered, so that the simulation model shown in FIG. 5C issuitable for the analysis of Maneuvering loads, in other words theaircraft is maneuvered in its entirety.

In the special case shown in FIG. 5D, the integral model is suitable forthe analysis of non-linear gusts of wind, safety and passenger comfort.

For the special case shown in FIG. 5E, the integral simulation model issuitable for the analysis of the system dynamics of the on-board system.

FIG. 2 shows an embodiment of a calculation system 1 according to theinvention for an aircraft 2, for example for an aeroplane. Sensors 3 areprovided on the aircraft or aeroplane 2. The sensors 3 are used todetect aeroelastic and flight-mechanical momenta of the aircraft 2.Furthermore, sensors are provided for detecting positions and movementsof control surfaces of the aircraft 2 and for detecting speeds of gustsof wind acting on the aircraft 2. Thus, the sensors 3 are controlsurface sensors, flight-mechanical sensors and aeroelastic sensors. Thesensors 3 for detecting flight-mechanical momenta of the aircraft 2 andfor detecting aeroelastic momenta of the aircraft have, for example,acceleration and pressure sensors. The sensors 3 for detectingflight-mechanical momenta can also measure deformations of parts of theaircraft 2. The calculation system 1 contains a calculation unit 4 whichcalculates characteristic quantities of passenger comfort and cabinsafety as well as momenta of the aircraft 2 as a function of the sensordata provided by the sensors 3 and a non-linear simulation model of theaircraft 2. In the embodiment shown in FIG. 2, this non-linearsimulation model is read out from a memory 5. In a possible embodiment,the calculation unit 4 has at least one microprocessor for implementingsimulation software for the integral simulation model. In a possibleembodiment, the non-linear simulation model is automatically adaptedusing the sensor data provided by the sensors 3 and rewritten in thememory 5.

In a possible embodiment, the calculation unit 4 is located in theaircraft 2 and receives the sensor data via an internal data bus fromthe sensors 3. In an alternative embodiment, the calculation unit 4 isnot located in the aircraft 2, but receives the sensor data via awireless air interface from the sensors 3. In this case, the calculationunit 4 can be located in a ground station, for example.

The calculation system 1 according to the invention, as shown in FIG. 2,also contains an input unit 6 for inputting parameters of the simulationmodel for the aircraft 2. In the embodiment shown in FIG. 2, thecharacteristic quantities and momenta calculated by the calculation unit4 are output via an output unit 7. The output unit 7 is, for example, anindication means or a display. The input unit 6 is, for example, akeyboard for inputting data. The input unit 6 and the output unit 7together form a user interface. This user interface can be used, forexample, by an engineer for aircraft design optimisation.

In the embodiment shown in FIG. 2, the characteristic quantities andmomenta calculated by the calculation unit 4 are coupled back to acomparison unit 8 in which a difference between precalculated quantitiesand quantities ascertained from the tests is calculated. Theprecalculated quantities can be input, for example, via a control unit 9and are compared with the simulated characteristic quantities. Theaircraft 2 is then controlled as a function of the difference orvariation between the precalculated and the simulated characteristicquantities.

The calculation unit 1 according to the invention allows the integraldynamic calculation of loads, momenta of the aeroelastics, flightmechanics and thus makes it possible for flight telemetry anddevelopment engineers to determine, with presettable accuracy, temporalcourses of all loads affecting the aircraft 2, and aeroelastic andflight-mechanical momenta using sensor data and to compare them with themeasured sensor data. This allows targeted aircraft design optimisation.

Furthermore, the calculation system 1 according to the invention, asshown in FIG. 2, is suitable for implementing targeted pilot training.For example, a pilot can compare control surface input data andresulting comfort, safety and loading characteristics of the aircraft 2.In this manner, airline pilots, test pilots and simulator pilots can betrained to avoid peak loads in dangerous flight situations or flyingmaneuvers, to reduce fatigue loads, to prevent critical vibrationstates, to reduce high accelerations in the entire cabin area of theaircraft and to increase passenger and crew safety and comfort.

This can also reduce operating costs for the customer and operator ofthe aircraft 2 and production costs of the aircraft 2. At the same time,the quality of the aircraft 2 is improved in respect of comfort, safetyand emission characteristics.

FIG. 3 shows a further embodiment of the calculation system 1 accordingto the invention for an aircraft 2. In the embodiment shown in FIG. 3,the aircraft 2 has what is known as an on-board system 10, it beingpossible for preferably different operating modes to be set. Theon-board system of the aircraft 2 is controlled automatically as afunction of the characteristic quantities and momenta calculated by thecalculation unit 4 to minimise load forces and vibrations. In a possibleembodiment, the on-board system 10 of the aircraft 2 can be connected toor disconnected from different frequency ranges. In a possibleembodiment, the on-board system 10 has different masses which are fittedto parts of the aircraft and can be activated as a function of theoperating mode of the on-board system 10. The on-board system 10 is usedto improve comfort and cabin safety and also to reduce loads on parts ofthe aircraft 2.

In the calculation system 1 according to the invention, as shown inFIGS. 2 and 3, sensors 3 detect the temporal development of loads andmomenta of the aeroelastics, flight mechanics and on-board systems ofthe aircraft 2 and also detect the control surface input data and theeffect of gusts of wind on the aircraft 2. The calculation unit 4 whichcan be, for example, a computer calculates characteristic quantities ofpassenger comfort and momenta of the aircraft 2 by means of simulationsoftware and the read-in simulation model. Also running on thecalculation unit 4 are input software and output software for inputtingparameters of the simulation model and for outputting the calculatedquantities. The integrated configuration of the calculation system 1with the on-board system 10 leads to an improvement in passengercomfort, cabin safety, aeroelastic and vibration characteristics and toa reduction of loads. Using identification software, it is possible tophysically identify partial and complete models of a high dimensionalparameter space, i.e. of the partial models, the aerodynamics of thestructure etc., using the available sensor data. The simulation softwareand the identification software, together with the input software forthe sensor data and the input and output software for the userinterface, are integrated into a software system.

FIGS. 6A and 6B show diagrams of embodiments of the calculation system 1according to the invention. FIG. 6A shows, for validating thecalculation system, a completely measured transmission function of anaileron to the lateral load factor on the front fuselage of an aircraft.The transmission function I in FIG. 6A shows the case where no on-boardsystem 10 is used to improve passenger comfort. The on-board system 10is switched on in the case of transmission function II in FIG. 6A. Thus,the on-board system 10 which is switched on as a function of theselected operating signal increases the aeroelastic damping in afrequency range of 2 to 3 Hz, as shown in FIG. 6A. However, as a result,the vibration characteristics, passenger comfort and safety are improvedas well as the direct fuselage loads due to fuselage movement.

FIG. 6B shows in an exemplary manner the transmission functions,determined by the calculation system 1 according to the invention, of anaileron with respect to a lateral load factor on the front fuselage ofan aircraft 2. The lateral load factor describes the load on the frontfuselage, comfort and crew and passenger safety under the effect ofgusts of wind or during extreme flight maneuvers and describesaeroelastic and vibration characteristics. The transmission function IIIin FIG. 6B shows the case where an on-board system 10 is not used toreduce the loads to improve passenger comfort and safety. The on-boardsystem 10 is switched on for the transmission function IV in FIG. 6B.

FIGS. 7A and 7B show diagrams of further embodiments to illustrate thecalculation system 1 according to the invention and the method accordingto the invention for determining characteristic quantities of passengercomfort and momenta of an aircraft 2.

FIG. 7A shows the temporal development of a characteristic quantity, forexample a load in the form of a scaled bending moment of a load on anouter wing of an aircraft 2 in the case of a spiral curved flight. Inthis respect, a vertical load factor NZ on a centre of gravity of theaircraft 2 is increased from 1 g to 1.5 g. Curve I in FIG. 7A shows thetemporal course according to a conventional integral simulation modelwithout the use of the sensor-based calculation system 1 according tothe invention. Curve II in FIG. 7A shows a temporal course for asimulation model using the sensor-based calculation system 1 accordingto the invention. Curve III in FIG. 7A shows, for validation, anactually measured load on an outer wing of the aircraft 2. As can beseen from FIG. 7A, the actual measured curve III is reproduced veryeffectively by curve II with the sensor-based calculation system of theinvention; in other words the simulated curve is almost completely thesame as the actual measured curve.

FIG. 7B shows the corresponding development of the load in the form of ascaled bending moment as a function of a current value of the loadfactor, i.e. the transformation of the time (see x-axis in diagram 7A)into the load factor n_(z)=b_(z)/g (see equation (21)). The load factoris a characteristic quantity which also describes the comfort, theaeroelastics and the safety of the aircraft 2. The load factorre-characterises the acceleration at the centre of gravity of theaircraft 2. It can also be seen from FIG. 7B that the curve calculatedby the calculation system 1 according to the invention coincides wellwith the actual measured curve.

In a possible embodiment of the method according to the invention,proceeding from a starting model which, for example corresponds to curveI in FIGS. 7A and 7B, the non-linear simulation model is automaticallyadapted using the sensor data. In a possible embodiment, this adaptioncan be performed iteratively. In a possible embodiment, the non-linearsimulation model is adapted to or validated with the sensor datasupplied by the sensors 3 by means of a least square algorithm (LSA). Bymeans of the calculation system according to the invention, it ispossible to simulate characteristic quantities of loads, i.e. forces onparts of the aircraft, of passenger comfort, i.e. for exampleacceleration forces on passenger seats or vibrations, aeroelasticcharacteristic quantities, characteristic quantities of the on-boardsystem as well as characteristic quantities and momenta of the flightmechanics. An integral optimisation of different characteristicquantities can be achieved with the non-linear simulation model which isused. For example, an engineer can simultaneously optimisecharacteristic quantities of passenger comfort and aeroelasticcharacteristic quantities, while considering the loads of the on-boardsystem 10 and of the flight mechanics. For example, the accelerationforces acting on the passenger seats can be minimised, while at the sametime aeroelastic characteristic quantities are calculated for minimisingmaterial wear and for maximising flight safety. Therefore, the inventionprovides an integral sensor-based calculation system 1 for loads,characteristic quantities of passenger comfort and cabin safety and formomenta of the aeroelastics, the structural dynamics, the stationary andinstationary aerodynamics and the on-board systems of aircraft 2.

The invention claimed is:
 1. A system for control, monitoring, anddesign validation of an aircraft, comprising: (a) at least one sensorfor detecting at least one of aeroelastic and flight-mechanical momentaof the aircraft, positions and movements of control surfaces of theaircraft, and speeds of gusts of wind acting on the aircraft; (b) acalculation unit having at least one microprocessor, the calculationunit being configured for calculating characteristic quantities ofpassenger comfort and cabin safety and momenta of the aircraft from dataprovided by the at least one sensor using a stored non-linear simulationmodel; and (c) a control unit configured for controlling the aircraftand coupled to a comparison unit configured for calculating a differencebetween precalculated quantities and the quantities calculated by thecalculation unit.
 2. The system according to claim 1, wherein the storednon-linear simulation model of the aircraft is a dynamic model of lineardifferential equations which is expanded by an effectiveness matrix (F)which is multiplied by a non-linearity vector (g), reading as:M{umlaut over (x)}+D{dot over (x)}+Kx+Fg(x,{dot over (x)},p,t)=p, whereM is a mass matrix, D is a damping matrix, K is a rigidity matrix, x isa hyper-movement vector of the aircraft, g is the non-linearity vector,p is a hyper-input vector of the aircraft, and F is the effectivenessmatrix, wherein the effectiveness matrix describes non-linearcharacteristics of characteristic quantities which haveflight-mechanical characteristic quantities, characteristic quantitiesof an on-board system and characteristic quantities of the aeroelastics.3. The system according to claim 1, wherein the calculation unitmicroprocessor automatically adapts the stored non-linear simulationmodel using the sensor data provided by the sensors.
 4. The systemaccording to claim 1, wherein sensors are provided for detecting momentaof an on-board system of the aircraft.
 5. The system according to claim3, wherein the on-board system has at least one movable mass for dampingat least one part of the aircraft.
 6. The system according to claim 1,wherein the sensors for detecting flight-mechanical momenta of theaircraft measure deformations of at least one part of the aircraft. 7.The system according to claim 1, wherein the sensors for detectingflight-mechanical momenta of the aircraft and for detecting aeroelasticmomenta of the aircraft have at least one of acceleration and pressuresensors.
 8. The system according to claim 1, wherein the calculationunit is located in at least one of the aircraft and a ground stationreceiving the sensor data from the sensors of the aircraft via awireless air interface.
 9. The system according to claim 1, wherein thenon-linear simulation model of the aircraft is read out of a memory. 10.The system according to claim 1, wherein the calculation unit isconnected to an input unit for inputting parameters of the simulationmodel of the aircraft.
 11. The system according to claim 1, wherein thecalculation unit is connected to an output unit for outputting thecalculated characteristic quantities and momenta.
 12. The systemaccording to claim 4, wherein the on-board system of the aircraft iscontrolled automatically as a function of the characteristic quantitiesand momenta calculated by the calculation unit, minimizing load forcesand vibrations.
 13. The system according to claim 12, wherein theon-board system of the aircraft is at least disconnected from differentfrequency ranges.
 14. The system according to claim 12, wherein at leasta mass of an on-board system which is fitted to at least one part of theaircraft is activated as a function of an adjustable operating mode ofthe on-board system.
 15. A method for controlling an aircraft,comprising the following steps: (a) detecting aeroelastic andflight-mechanical momenta of the aircraft, of positions and movements ofcontrol surfaces of the aircraft, and speeds of gusts of wind acting onthe aircraft, to generate sensor data; (b) calculating characteristicquantities of passenger comfort and momenta of the aircraft as afunction of the generated sensor data and a stored non-linear simulationmodel of the aircraft; and (c) calculating a difference betweenprecalculated quantities and the quantities calculated by thecalculation unit, the aircraft being controlled as a function of thecalculated difference.
 16. A computer program product comprising programcommands that when executed by a processor of a computer control thecomputer to perform steps comprising: detecting aeroelastic andflight-mechanical momenta of an aircraft, positions and movements ofcontrol surfaces of the aircraft, and speeds of gusts of wind acting onthe aircraft, to generate sensor data; calculating characteristicquantities of passenger comfort and momenta of the aircraft as afunction of the generated sensor data and a stored non-linear simulationmodel of the aircraft, and calculating a difference betweenprecalculated quantities and the quantities calculated by thecalculation unit, the aircraft being controlled as a function of thecalculated difference.
 17. A data carrier comprising storing means forstoring a computer program product comprising program commands that whenexecuted by a processor of a computer control the computer to performsteps comprising: detecting aeroelastic and flight-mechanical momenta ofan aircraft, positions and movements of control surfaces of theaircraft, and speeds of gusts of wind acting on the aircraft, togenerate sensor data; calculating characteristic quantities of passengercomfort and momenta of the aircraft as a function of the generatedsensor data and a stored non-linear simulation model of the aircraft,and calculating a difference between precalculated quantities and thequantities calculated by the calculation unit, the aircraft beingcontrolled as a function of the calculated difference.